Clifford algebra bundles

More generally a real vector bundle $E \to M$ is a Euclidean bundle if, with $\sE = \Ga(M, E^\bC)$, there is a symmetric $A$-bilinear form $g \: \sE \x \sE \to A = C(M)$ such that

1. $g(s,t) \in C(M;\bR)$ when $s,t$ lie in $\Ga(M, E)$ --the real sections;
2. $g(s,s) \geq 0$ for $s \in \Ga(M, E)$, with $g(s,s) = 0 \implies s = 0$.

By defining $\pairing{s}{t} := g(s^*,t)$, we get a hermitian pairing with values in $A$:

• $\pairing{s}{t}$ is $A$-linear in $t$;
• $\pairing{t}{s} = \overline{\pairing{s}{t}} \in A$;
• $\pairing{s}{s} \geq 0$, with $\pairing{s}{s} = 0 \implies s = 0$ in $\sE$;
• $\pairing{s}{ta} = \pairing{s}{t} a$ for all $s,t \in \sE$ and $a \in A$.

These properties make $\sE$ a (right) $C^*$-module over $A$, with $C^*$-norm given by

$$\Vert s\Vert _{\sE} := \sqrt{\Vert\pairing{s}{s}\Vert _A} \word{for} s \in \sE.$$

For each $x \in M$, we can form $\bCl(E_x) := \Cl(E_x,g_x) \ox_\bR \bC$. Using the linear isomorphisms $\sg_x \: \bCl(E_x) \to (\La^\8 E_x)^\bC$, we see that these are fibres of a vector bundle $\bCl(E) \to M$, isomorphic to $(\La^\8 E)^\bC \to M$ as $\bC$-vector bundles (but not as algebras!). Under $(\ka\la)(x) := \ka(x)\la(x)$, the sections of $\bCl(E)$ also form an algebra $\Ga(M, \bCl(E))$. It has an $A$-valued pairing

$$\pairing{\ka}{\la} \: x \mapsto \tau(\ka(x)^* \la(x)).$$

By defining $\Vert\ka\Vert := \sup_{x\in M} \Vert\ka(x)\Vert _{\bCl(E_x)}$, this becomes a $C^*$-algebra.